New treatments of density fluctuations and recurrence times for re-estimating Zermelo’s paradox

What is the probability that all the gas in a box accumulates in the same half of this box? Though amusing, this question underlies the fundamental problem of density fluctuations at equilibrium, which has profound implementations in many physical fields. The currently accepted solutions are derived from the studies of Brownian motion by Smoluchowski, but they are not appropriate for the directly colliding particles of gases. Two alternative theories are proposed here using self-regulatory Bernoulli distributions, which incorporate roles for crowding and pressure in counteracting density fluctuations. A quantum of space is first introduced to develop a mechanism of matter congestion holding for high densities. In a second mechanism valid in ordinary conditions, the influence of local pressure on the location of every particle is examined using classical laws of ideal gases. This approach reveals that a negative feedback results from the reciprocal influences between individual particles and the population of particles, which strongly reduces the probability of atypical microstates. Finally, a thermodynamic quantum of time is defined to compare the recurrence times of improbable macrostates predicted through these different approaches.